scholarly journals Theorems Relating to Quadratic Forms and their Discriminant Matrices

1953 ◽  
Vol 10 (1) ◽  
pp. 13-15
Author(s):  
S. Vajda
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Degrees Of Freedom ◽  
Normal Population ◽  
Royal Statistical Society ◽  
Unit Variance ◽  
The Matrix ◽  
Chi Squared ◽  
Image Position ◽  
Latent Roots

In a paper read before the Research Branch of the Royal Statistical Society (Ref. 1, p. 150) the following case was considered:Let the expression be given; introduce, for c, a linear form in and obtainIf the yi are sample values from a normal population with unit variance, then it is known (Ref. 2) that (1) is distributed as where zi varies as chi-squared with one degree of freedom and the li are the latent roots of the matrix of the quadratic form. If these latent roots are f times unity and n—f times zero, then this reduces to a chi-squared distribution with f degrees of freedom.

The inhomogeneous minimum of quadratic forms of signature ± 1

1981 ◽  
Vol 89 (2) ◽  
pp. 225-235 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

Let Qr be a real indefinite quadratic form in r variables of determinant D ≠ 0 and of type (r1, r2), 0 < r1 < r, r = r1 + r2, S = r1 − r2 being the signature of Qr. It is known (e.g. Blaney (3)) that, given any real numbers c1, c2,…, cr, there exists a constant C depending only on r and s such that the inequalityhas a solution in integers x1, x2, …, xr.

Pearson’s chi-squared statistics: approximation theory and beyond

Biometrika ◽  
2019 ◽  
Vol 106 (3) ◽  
pp. 716-723
Author(s):  
Mengyu Xu ◽  
Danna Zhang ◽  
Wei Biao Wu
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Approximation Theory ◽  
Quadratic Forms ◽  
Degrees Of Freedom ◽  
Goodness Of Fit ◽  
High Dimensional ◽  
Goodness Of Fit Test ◽  
Chi Squared

Summary We establish an approximation theory for Pearson’s chi-squared statistics in situations where the number of cells is large, by using a high-dimensional central limit theorem for quadratic forms of random vectors. Our high-dimensional central limit theorem is proved under Lyapunov-type conditions that involve a delicate interplay between the dimension, the sample size, and the moment conditions. We propose a modified chi-squared statistic and introduce an adjusted degrees of freedom. A simulation study shows that the modified statistic outperforms Pearson’s chi-squared statistic in terms of both size accuracy and power. Our procedure is applied to the construction of a goodness-of-fit test for Rutherford’s alpha-particle data.

scholarly journals Positive values of inhomogeneous quinary quadratic forms of type (4,1)

1981 ◽  
Vol 31 (2) ◽  
pp. 175-188 ◽  
Author(s):  
R. J. Hans-Gill ◽  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Image Position

AbstractHere it is proved that if Q(x, y, z, t, u) is a real indefinite quinary quadratic form of type (4,1) and determinant D, then given any real numbers x0, y0, z0, t0, u0 there exist integers x, y, z, t, u such thatAll critical forms are also obtained.

Integral Bases for Quadratic Forms

1963 ◽  
Vol 15 ◽  
pp. 412-421 ◽  
Author(s):  
J. H. H. Chalk
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Integer Variables ◽  
Inhomogeneous Minimum ◽  
Integral Bases ◽  
Indefinite Quadratic Form ◽  
Homogeneous Minimum ◽  
Indefinite Quadratic ◽  
Image Position

Letbe an indefinite quadratic form in the integer variables x1, . . . , xn with real coefficients of determinant D = ||ars||(n) ≠ 0. The homogeneous minimum MH(Qn) and the inhomogeneous minimum MI(Qn) of Qn(x) are defined as follows :

Inhomogeneous minimum of indefinite quadratic forms in six variables: A conjecture of Watson

1983 ◽  
Vol 94 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Famous Conjecture ◽  
Inhomogeneous Minimum ◽  
Indefinite Quadratic Form ◽  
History Of ◽  
Indefinite Quadratic ◽  
Image Position

The famous conjecture of Watson(11) on the minima of indefinite quadratic forms in n variables has been proved for n ≤ 5, n ≥ 21 and for signatures 0 and ± 1. For the details and history of the conjecture the reader is referred to the author's paper(8). In the succeeding paper (9), we prove Watson's conjecture for signature ± 2 and ± 3 and for all n. Thus only one case for n = 6 (i.e. forms of type (1, 5) or (5, 1)) remains to he proved which we do here; thereby completing the case n = 6. This result is also used in (9) for proving the conjecture for all quadratic forms of signature ± 4. More precisely, here we prove:Theorem 1. Let Q6(x1, …, x6) be a real indefinite quadratic form in six variables of determinant D ( < 0) and of type (5, 1) or (1, 5). Then given any real numbers ci, 1 ≤ i ≤ 6, there exist integers x1,…, x6such that

scholarly journals On indefinite ternary quadratic forms

1974 ◽  
Vol 18 (4) ◽  
pp. 388-401 ◽  
Author(s):  
R. T. Worley
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Ternary Quadratic Forms ◽  
Indefinite Ternary Quadratic Form ◽  
Image Position

In a paper [1] with the same title Barnes has shown that if Q(x, y, z) is an indefinite ternary quadratic form of determinant d ≠ 0 then there exist integers x1, y1, z1, x2,···z3 satisfying for which Furthermore, unless Q is equivalent to a multiple of or two other forms Q2, Q3 then the constant ⅔ in (1.2) can be replaced by 1/2.2. For Q1 equality is needed on at least one side of (1.2) while for Q2, Q3 the constant ⅔ can be reduced to 12/25 but no further.

The positive values of inhomogeneous ternary quadratic forms

1961 ◽  
Vol 2 (2) ◽  
pp. 127-132 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Equality Sign ◽  
Ternary Quadratic Forms ◽  
Positive Multiple ◽  
Indefinite Ternary Quadratic Form ◽  
Image Position

Let f(x, y, z) be an indefinite ternary quadratic form of signature (2, 1) and determinant d ≠ 0. Davenport [3] has shown that there exist integral x, y, z with, the equality sign being necessary if and only if f is a positive multiple of f1(x, y, z) = x2 + yz.

scholarly journals On a Normal Form of the Orthogonal Transformation I

1958 ◽  
Vol 1 (1) ◽  
pp. 31-39 ◽  
Author(s):  
Hans Zassenhaus
Keyword(s):  
Normal Form ◽  
Quadratic Form ◽  
Lorentz Transformation ◽  
Orthogonal Transformation ◽  
Complex Conjugate ◽  
Linear Transformations ◽  
Characteristic Roots ◽  
The Matrix ◽  
Real Characteristic ◽  
Image Position

At the Edmonton Meeting of the Canadian Mathematical Congress E. Wigner asked me whether one knew something about the distribution of the characteristic roots of the linear transformations that leave invariant the quadratic form t2+x2-y2-z2, just as one knows that a Lorentz transformation has two complex conjugate characteristic roots and two real characteristic roots that are either inverse to one another or the numbers 1 and -1.In this paper an answer to E. Wigner’s question will be obtained.We are concerned with the pairs of matrices (X,A) with coefficients in a field of reference F such that the condition0.1is satisfied, where XT = (ξki) is the transpose of the matrix X = (ξki). It follows that both matrices are quadratic of the same degree d.

scholarly journals III. On the orders and genera of quadratic forms containing more than three indeterminates.—Second notice

1868 ◽  
Vol 16 ◽  
pp. 197-208 ◽  
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Royal Society ◽  
Present Communication ◽  
The Matrix ◽  
Primitive Quadratic Form ◽  
Definition Of ◽  
Greatest Common Divisors

The principles upon which quadratic forms are distributed into orders and genera have been indicated in a former notice (Proceedings of the Royal Society, vol. xiii. p. 199). Some further results relating to the same subject are contained in the present communication. I. The Definition of the Orders and Genera. Retaining, with some exceptions to which we shall now direct attention, the notation and nomenclature of the former notice, we represent by f 1 a primitive quadratic form containing nindeterminates, of which the matrix is || A n x n i, j ; by f 2 , f 3 , . . . f n -1 , the fundamental concomitants o f 1 , of which the last is the contravariant. The matrices of these concomitants are the matrices derived from the matrix of f 1 , so that the first coefficients of f 2 , f 3 , .. . f n -1 , are respectively the determinants |A 2 x 2 i, j |, | · A 3 x 3 i, j |,... |A n -1 x n -1 i, j |, taken with their proper signs. The discriminant of f 1 , i. e. the determinant of the matrix |A n x n i, j |, which is supposed to be different from zero, and which is to be taken with its proper sign, is represented by ∇ n . The greatest common divisors of the minors of the orders n - 1, n - 2, . . . 2, 1 in the same matrix are denoted by ∇ n -1 , ∇ n -2 , ∇ 2 , ∇ 1 , of which the last is a unit; we shall presently attribute signs to each of these greatest common divisors.

Note on Integers Representable by Binary Quadratic Forms

1975 ◽  
Vol 18 (1) ◽  
pp. 123-125 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Error Term ◽  
Asymptotic Estimate ◽  
Complex Analysis ◽  
Binary Quadratic Form ◽  
Binary Quadratic Forms ◽  
Transcendental Method ◽  
Image Position ◽  
Positive Integers

Let B be the set of positive integers prime to d which are representable by some primitive, positive, integral binary quadratic form of discriminant d. It is the purpose of this note to show that the following asymptotic estimate for the number of integers in B less than or equal to x can be proved using only elementary arguments:(1)where c1 is the positive constant given in (17) below. (Using the deeper methods of complex analysis James [2] has proved this result with the error term ((log x)-1/2) replacing ((log log x)-1). Heupel [1] using a transcendental method as in James [2] improved this to ((log x)-1).)

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